On Krylov Complexity as a Probe of the Quantum Mpemba Effect

TL;DR

Frames the quantum Mpemba effect (QME) and motivates Krylov state complexity as a transparent probe of relaxation in many-body spin chains, highlighting how global symmetries such as $U(1)$ shape diagnostic signals. It outlines the Lanczos-based Krylov formalism and decompositions into symmetric ($\mathcal{C}_S(t)$) and asymmetric ($\mathcal{C}_A(t)$) parts to probe intra- and inter-sector dynamics. In symmetry-broken models, the total complexity $\mathcal{C}(t)$ exhibits Mpemba-like inversions, with early-time quadratic growth and late-time saturation tied to energy-space delocalization; tilted ferromagnetic states show crossings that are absent for tilted Néel states. In $U(1)$-symmetric systems, the symmetric component (and its shifted form $\tilde{\mathcal{C}}_S$) provides robust Mpemba diagnostics across multiple spin-chain variants, clarifying the role of coherence between sectors. The results support symmetry-resolved Krylov diagnostics as a universal framework for diagnosing anomalous relaxation and suggest extensions to infinite-dimensional settings such as quantum field theories.

Abstract

We investigate Krylov state complexity as a probe of the quantum Mpemba effect in quantum spin chains. For models without global $U(1)$ symmetry, Krylov complexity exhibits clear Mpemba-like crossings, consistent with conventional diagnostics such as the trace distance, while offering a complementary interpretation in terms of Hilbert-space exploration and dynamical delocalization. In $U(1)$-symmetric systems, we confirm that the recently proposed symmetric component of Krylov complexity serves as a robust and reliable indicator of the QME, capturing anomalous relaxation even in cases where the total complexity fails to do so.

Figures (8)

• Figure 1: Early-time growth of Krylov complexity for different tilt angles $\theta$ in next-nearest-neighbor XXZ Hamiltonian (Right) and mixed-field Ising chain (Left). Strongly tilted states initially grow more slowly but eventually overtake weakly tilted ones, a signature of Mpemba-like behavior.
• Figure 2: Krylov complexity of TFS (Left) and TNS (Right) for different values of $\theta$.
• Figure 3: Infinite-time averages of Krylov complexity of TFS (Left) and TNS (Right) for different values of $\theta$ and $N=9$.
• Figure 4: Infinite-time average of complexity (left) and logarithm of the inverse participation ratio (right) for general initial states given in \ref{['initial']}, with $N=9$. Due to the symmetry under $\phi \rightarrow 2\pi - \phi$, we show the results only for $0 \leq \phi \leq \pi$.
• Figure 5: Difference in Krylov complexity (left) and its symmetric component (right) between initial states with tilt parameters $\theta_1 = 0.15$ and $\theta_2 = 0.2$, for the TFS in the model given by \ref{['eq:XXz2']}. System size is $N=10$.